Lecture 20: Fourier Transform, The Wave Equation, and Weak Solutions

In this lecture we defined the Fourier transform of a tempered distribution and proved that it satisfies the same good properties as the Fourier transform on Schwartz functions. We then used it to solve the IVP for the wave equation:

$$({\partial }_t-{\mathrm{\Delta }}_x)u=0u(0,x)=\varphi (x){\partial }_{t}u(0,x)=\psi (x).$$

The development followed Andras Vasy's notes http://math.stanford.edu/~andras/172-5.pdf (see page 8 onwards) very closely. The point is that even though the solution

$u(x,t)$ is eventually a

$C^{\mathrm{\infty }}$ function, it is easier to pass through the space of tempered distributions to find this solution.